<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">x</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:14_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><a NAME="E3:14_1"/><b>then </b>
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:14_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T19">WAYBEL_0:19</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:14"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D1">WAYBEL_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:14_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:14_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/><b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i/>;<br/><a NAME="E4:14_2"/>
<font color="Maroon">y'</font> is <a href="waybel_3.html#V1">compact</a>
 <b>by </b><i>, <a class="ref" href="waybel_8.html#D1">WAYBEL_8:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:14_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">x'</font> = <font color="Maroon">x'</font>;<br/><b>thus </b><a NAME="E8:14_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/><b>thus </b><a NAME="E9:14_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E5:14"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T16">FUNCT_2:16</a></i>;<br/>
<a NAME="E6:14"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> iff <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>reconsider </b><font color="Maroon">fx</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fy</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/>
<a NAME="E3:14_3"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
;<br/>
<a NAME="E4:14_3"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<b>thus </b><a NAME="E5:14_3"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_3_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:14_3_1"/><b>then </b>
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T21">WAYBEL_0:21</a></i>;<br/>
<a NAME="E3:14_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:14_3_1"/><b>then </b>
<font color="Maroon">fx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">fy</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:14_3_1"/><b>then </b>
<font color="Maroon">fx</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">fy</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<b>hence </b><a NAME="E6:14_3_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:14_3"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> implies <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_3_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:14_3_2"/><b>then </b>
<font color="Maroon">fx</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">fy</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/>
<a NAME="E3:14_3_2"/><b>then </b>
<font color="Maroon">fx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">fy</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<a NAME="E4:14_3_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:14_3_2"/><b>then </b>
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:14_3_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:14"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>


</div>
